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Math Help - Prove: If H is a finite group of even cardinality, the H contains an element h of ord

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    Prove: If H is a finite group of even cardinality, the H contains an element h of ord

    Prove: If H is a finite group of even cardinality, the H contains an element h of order 2.
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    Quote Originally Posted by apple2009 View Post
    Prove: If H is a finite group of even cardinality, the H contains an element h of order 2.
    Problem: Suppose \left|H\right| is even. Prove that at least one element of H is of order two.

    Proof: Suppose that H contained no elements of order two. We can see then that a\ne a^{-1} for all nontrivial elements of H. Therefore elements of this nature come in distinct pairs ( a,a^{-1}) adding up all these elements will then give an even number. Then noting that since no nontrivial element of H had order two we can see that the total number of elements in H is the number of all elements whose order isn't two and the identity element. But this is an even number plus one, thus odd which is of course a contradiction.
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